Find the sums given below:

(i) 34 + 32 + 30 + …. + 10

(ii) -5 + (-8) + (-11) + …. + (-230)

(i) The given numbers form an A.P. with a = 34 and d = 32 - 34 = -2.

Let nth term be 10,
as, a_n = a + (n - 1)d

∴ 10 = 34 + (n - 1)(-2)
⇒ 10 = 34 - 2n + 2
⇒ 10 = 36 - 2n
⇒ 2n = 36 - 10
⇒ 2n = 26
⇒ n = 13.

$\text{Sum } = \dfrac{n}{2}[2a + (n - 1)d] \\[1em] \therefore \text{ Sum } = \dfrac{13}{2}[2 \times 34 + (13 - 1)(-2)] \\[1em] = \dfrac{13}{2}[68 + 12(-2)] \\[1em] = \dfrac{13}{2}[68 - 24] \\[1em] = \dfrac{13}{2} \times 44 \\[1em] = 13 \times 22 \\[1em] = 286.$

Hence, the sum of the series 34 + 32 + 30 + …. + 10 is 286.

(ii) The given numbers form an A.P. with a = -5 and d = -8 - (-5) = -3.

Let nth term be -230, then as, a_n = a + (n - 1)d

∴ -230 = -5 + (n - 1)(-3)
⇒ -230 = -5 -3n + 3
⇒ -230 = -2 - 3n
⇒ -230 + 2 = -3n
⇒ -228 = -3n
⇒ 3n = 228
⇒ n = 76.

$\text{Sum } = \dfrac{n}{2}[2a + (n - 1)d] \\[1 em] \therefore \text{Sum } = \dfrac{76}{2}[2 \times -5 + (76 - 1)(-3)] \\[1em] = 38[-10 + 75(-3)] \\[1em] = 38[-10 - 225] \\[1em] = 38 \times -235 \\[1em] = -8930.$

Hence, the sum of the series -5 + (-8) + (-11) + …. + (-230) is -8930.